Mathematics – Statistics Theory
Scientific paper
2010-08-03
Mathematics
Statistics Theory
Scientific paper
Consider the standard Gaussian linear regression model $Y=X\theta+\epsilon$, where $Y\in R^n$ is a response vector and $ X\in R^{n*p}$ is a design matrix. Numerous work have been devoted to building efficient estimators of $\theta$ when $p$ is much larger than $n$. In such a situation, a classical approach amounts to assume that $\theta_0$ is approximately sparse. This paper studies the minimax risks of estimation and testing over classes of $k$-sparse vectors $\theta$. These bounds shed light on the limitations due to high-dimensionality. The results encompass the problem of prediction (estimation of $X\theta$), the inverse problem (estimation of $\theta_0$) and linear testing (testing $X\theta=0$). Interestingly, an elbow effect occurs when the number of variables $k\log(p/k)$ becomes large compared to $n$. Indeed, the minimax risks and hypothesis separation distances blow up in this ultra-high dimensional setting. We also prove that even dimension reduction techniques cannot provide satisfying results in an ultra-high dimensional setting. Moreover, we compute the minimax risks when the variance of the noise is unknown. The knowledge of this variance is shown to play a significant role in the optimal rates of estimation and testing. All these minimax bounds provide a characterization of statistical problems that are so difficult so that no procedure can provide satisfying results.
No associations
LandOfFree
Minimax risks for sparse regressions: Ultra-high-dimensional phenomenons does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Minimax risks for sparse regressions: Ultra-high-dimensional phenomenons, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Minimax risks for sparse regressions: Ultra-high-dimensional phenomenons will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-614858